2Vectors

IA Vectors and Matrices

2.5 Scalar triple product

Definition (Scalar triple product). The scalar triple product is defined as

[a, b, c] = a · (b × c).

Proposition.

If a parallelepiped has sides represented by vectors

a, b, c

that

form a right-handed system, then the volume of the parallelepiped is given by

[a, b, c].

b

c

a

Proof.

The area of the base of the parallelepiped is given by

|b||c|sin θ

=

|b × c|

.

Thus the volume=

|b × c||a|cos φ

=

|a · (b × c)|

, where

φ

is the angle between

a

and the normal to

b

and

c

. However, since

a, b, c

form a right-handed system,

we have a · (b × c) ≥ 0. Therefore the volume is a · (b × c).

Since the order of a, b, c doesn’t affect the volume, we know that

[a, b, c] = [b, c, a] = [c, a, b] = −[b, a, c] = −[a, c, b] = −[c, b, a].

Theorem. a × (b + c) = a × b + a × c.

Proof. Let d = a ×(b + c) −a ×b −a ×c. We have

d · d = d · [a × (b + c)] − d · (a × b) − d · (a × c)

= (b + c) · (d × a) − b · (d × a) − c · (d × a)

= 0

Thus d = 0.